Proof: Proposition 6 in [1]} ([1]}) proves that \(\vert \vert \vert {\mathbf {X}}^{\lceil M \rceil }_i-{\mathbf {X}}_i\vert \vert \vert \)  converges to 0 in probability. Proposition 7 in [1]} ([1]}) proves that \(\vert \vert \vert {\mathbf {X}}^{\lceil M \rceil }_i-\widehat{\mathbf {X}}_i\vert \vert \vert =O_p(M_{\rm max}\triangle _M r_n)\) , where \(M_{\rm max}=\max \limits _{j=1,\ldots ,p}M_j\) , \(r_n= n^{−1}h^{-2}\)  in the case of measurement error or irregularly sampled data for a certain bandwidth \(h\) . Since \(\mathbf {\mu }\)  is estimated marginally to \(p\)  variables, the convergence rate of \(\widehat{\mu }^{(j)}(t^{(j)})-{\mu }^{(j)}(t^{(j)})\)  ([5]} [5]}) still holds, \(\sup \limits _{t^{(j)} \in \mathcal {T}_j}|\widehat{\mu }^{(j)}(t^{(j)})-{\mu }^{(j)}(t^{(j)})|=O_P(\frac{1}{\sqrt{n}h_{\mu ^{(j)}}})\) , and we name the bandwidth \(h_{min,\mu }:=\min \limits _{j}\lbrace h_{\mu ^{(j)}}\rbrace \) . Therefore \(\sup \limits _{\mathbf {t} \in \mathcal {T}}\Vert \widehat{\mathbf {\mu }}(\mathbf {t})-{\mathbf {\mu }}(\mathbf {t})\Vert =O_P(\frac{1}{\sqrt{n}h_{min,\mu }})\) . Therefore, \(\vert \vert \vert \widehat{\mathbf {X}}_i-{\mathbf {X}}_i\vert \vert \vert \)  converges to 0 in probability for \(M_1,\ldots , M_p \rightarrow \infty \) . \(\Box \)
